Three years ago, Jeb Bush left the Florida governor’s office with a legacy of having brought sweeping changes to his state’s education system, through hard-edged policies that gave parents and students more choices and demanded more of schools.

Today, that legacy seems poised to grow—and well beyond Florida.

In state capitals across the country, numerous lawmakers and officeholders say they are determined to follow the ambitious and often polarizing education blueprint fashioned by Mr. Bush, a Republican, during his two terms as Florida’s governor.